DEPARTMENT OF ELECTRICAL & ELECTRONICS ENGINEERING
COURSE: ELECTRICAL MEASUREMENTS
BRANCH: Electrical and Electronic Engineering
CLASS: III/I Sem.
YEAR: 2013-14
LECTURE NOTES
SHRI VISHNU ENGINEERING COLLEGE FOR WOMEN
VISHNUPUR, BHIMAVARAM - 534202
CONTENTS
UNIT NO. TITLE
PAGE NO.
UNIT I Measuring Instruments………………………………………………
02-11
UNIT II Instrument transformers and Special Meters................................
12-19
UNIT III Measurement of Power and Energy………………………………
20-27
UNIT IV Potentiometers……………………………………………………
28-33
UNITV Resistance Measurement……………………………………………
34-42
UNIT VI AC Bridges………………………………………………………
43-47
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UNIT-I
Measuring Instruments
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UNIT-I
1.1. METHODS OF MEASUREMENTS AND MEASURING INSTRUMENTS
The measurement methods can be analog or digital methods, deflection or null methods,
active or passive methods, direct or indirect methods and absolute or secondary methods.
Measurement generally involves an instrument as a physical means of determining an
unknown quantity or a variable called the parameter. The instrument is a means for
determining the value or magnitude of the measurand. The instruments can also be
divided into separate classes according to several criteria as, analog or digital
instruments, deflection or null type instruments, power operated (active) or self
generating (passive) instruments, contacting or non-contacting instruments, mechanical
or electrical instruments and monitoring or control instruments.
Signals which vary continuously with the change in the measurand are analog
signals and the devices producing them are analog instruments. The deflection
type dynamometer type wattmeter is a good example of an analog instrument. As
the input value changes, the moving system or pointer exhibits a smooth
continuous motion. The signals which vary in discrete steps and have only finite
number of values in any given range are digital signals and the associated devices
are digital instruments. A digital instrument has an output varying in discrete
steps. An electronic counter is an example of a digital instrument.
If the quantity is to be directly measured, then deflection methods are used. For
e.g., ammeter, voltmeter, etc. acting as meters indicating the value of the
measurand by the deflection of a pointer over a graduated and calibrated scale.
Alternatively, if the value is measured based on the null balance conditions, then
it is a null method. Null methods are used only to detect the null condition of a
measurand through a given path or circuit. AC/DC Bridge measurements for
measurement of resistance, inductance, capacitance, frequency, etc. are null
methods. They involve balance detection by using null detectors, such as,
Galvanometer, Vibration Galvanometers and Head Phones. Null instruments are
more accurate than the deflection instruments.
If the output of the instrument is entirely produced by the measurand, then it is an
active instrument. These are the power operated instruments requiring some
source of auxiliary power for their operation such as compressed air, electricity
and hydraulic supply. On the other hand, if the measurand modulates the
magnitude of some external power source, then it is a passive instrument. Passive
instruments are self generating instruments where the energy requirements are
met entirely from the input signal.
The direct methods involve measuring the measurand by comparison against its
own standard. They are very common for measurement of physical quantities
such as length, mass and time. They are less sensitive and inaccurate since they
involve human operators. Thus direct methods are not usually preferred. On the
other hand, indirect methods use measuring systems, which are the systems
having a transducer to convert the measurand into its analogous form. This
converted signal is processed, fed to the end devices to obtain the results.
Absolute methods give the magnitude of the quantity under measurement in terms
of the physical constants of the instrument. They do not require calibration. They
are used only for calibration of other instruments. For e.g., Tangent
Galvanometer, Rayleigh's current balance and potentiometers. Secondary methods
are so constructed that the desired quantity is measured only by observing the
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output of the instrument, which needs to be calibrated. Thus, they measure the
quantity in terms of their deflection, for which they are already calibrated. These
are the ones which are the most commonly used. For e.g., Voltmeters,
Thermometer, Pressure gauge, etc. Secondary methods work on either Analog
mode or Digital mode and hence lead to analog or digital methods.
Contacting type instruments are those which are kept in the measuring medium
itself. For e.g., clinical thermometer. A non-contacting or proximity type
instrument measures the desired input even though it is not in close contact with
the measuring medium. For e.g., optical pyrometer measuring the temperature of
a blast furnace, variable reluctance tachometer measuring the speed of a rotating
body, etc.
Mechanical instruments are very reliable for static conditions. Their parts are very
bulky, rigid and have a heavy mass. Hence they cannot respond rapidly to
measurements of dynamic and transient conditions. Besides, many of them are the
potential sources of noise. On the other hand, electrical instruments are very rapid
in response. However, their operating mechanism normally depends on a
mechanical meter movement as an indicating device.
Monitoring instruments are useful for monitoring functions. If their output is in a
form that can be directly included as part of an automatic control system, then
they become control instruments.
1.2. Requirements of Instruments
A measuring instrument should possess some of the following important characteristic
features:
• It should have a very high instrument efficiency which is the ratio of the quantity
being measured to the power utilized by the instrument at full scale.
• It should have a high sensitivity which is the ratio of the magnitude of the output
signal to the magnitude of the quantity being measured. The inverse of this ratio is
the Inverse Sensitivity or Deflection Factor.
• The output of the instrument should be linearly proportional to the input. In such
cases, the scale will be uniform and hence easier to calibrate.
• It should have a very low threshold which is the minimum value of below which
the change in output cannot be detected by the instrument.
• It should have lowest dead time which is the minimum time required for the
instrument to respond to a change in the quantity being measured.
• It should virtually have no dead zone where dead zone is the largest change of the
input quantity for which there is no output of the instrument. Dead zone is caused
due to friction, hysterisis, backlash, etc.
• It should have perfect reproducibility (precision) which is specified in terms of
the scale readings over a given period of time. This is different from repeatability
which is defined as the variation of scale readings and is random in nature.
• It should not have any drift. For an instrument, perfect reproducibility means that
the instrument has no drift. This means that for a given input, the measured values
are constant and do not vary with time.
• It should have minimum noise. Noise is any signal that does not convey any
useful information. Noise is due to extraneous disturbances caused by many
sources such as stray fields, shocks and thermal stresses.
• It should be modifiable and properly priced.
All the indicating instruments require three important torques for their operation:
deflecting torque-Td, controlling torque-Tc and damping torque-TD.
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 The deflecting torque is responsible for the movement of the pointer in proportion
to the value of the measurand. It is provided by the different effects of electric
current on which the operation of the given instrument depends.
 The controlling torque is responsible for controlling the movements of the
pointer. It is very high at the null position of the pointer. When the pointer gets
deflected due to the deflecting torque exerted on it, the controlling torque
provides the retarding torque and the two torques are equal at the equilibrium
position of the 5
 pointer. The controlling torque is provided by spring or gravity control. This also
ensures the zero setting of the pointer when the deflecting torque is absent.
 The damping torque provides the forces required to damp out the oscillatory
movements of the pointer, if any, due to the inertia of the system. The damping
torque is provided by eddy current, air friction or fluid friction methods.
1.3. Classification of Instruments
 Various effects of electric current are made use of by electrical instruments for
their operation, such as, magnetic effect, heating effect, chemical effect,
electrostatic effect and electromagnetic induction effect.
 Since the instruments using magnetic effect, such as, ammeter, voltmeter and
wattmeter, are simpler, cheaper, and could be commonly adopted on both ac and
dc circuits, they are more common.
 The instruments using heating effect are not much used due to their cost and
comparable inaccuracies.
 The instruments using chemical effect, such as, energy meters, are also not very
common, due to their cost and complications involved.
 The electrostatic effect is made use of in voltmeters, both on ac and dc. Though
expensive, the electrostatic voltmeters are very useful for high voltage
measurements.
 Electromagnetic induction forms the basis for many instruments, though on only
AC. The common meters using the electromagnetic induction effect of current
are: ammeter, voltmeter, wattmeter and energy meter. Of these, the wattmeter and
energy meter are more common, although they are costlier.
 Instruments can also be classified in a more general way as indicating
instruments, recording instruments and integrating instruments.
 Indicating instruments provide information about the measuring quantity in terms
of the deflection of a pointer over a pre-calibrated scale. For e.g., ammeter,
voltmeter, wattmeter, etc.
 Recording instruments make a record of the unknown quantity, usually on a
paper, against time or any other variable. For e.g., strip-chart temperature
recorder. The recording instruments are usually used in power houses and
substations, where a continuous record of the current, voltage, power or energy is
required.
 Integrating instruments always record the unknown quantity in an integrated
manner indicating the total cumulative value of the measurand at any instant of
time. For e.g., energy meters and ampere-hour meters.
 General Theory Permanent Magnet Moving Coil (PMMC) Instruments
 The general theory of moving-coil instruments may be dealt with considering a
rectangular coil of turns, free to rotate about a vertical axis. N
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Fig. shows the basic construction of a PMMC instrument. A moving coil
instrument consists basically of a permanent magnet to provide a magnetic field
and a small lightweight coil is wound on a rectangular soft iron core that is free to
rotate around
its vertical axis. When a current is passed through the coil windings, a torque is
developed on the coil by the interaction of the magnetic field and the field set up
by the current in the coil. The aluminum pointer attached to rotating coil and the
pointer moves around the calibrated scale indicates the deflection of the coil.
To reduce parallax error a mirror is usually placed along with the scale. A
balance weight is also attached to the pointer to counteract its weight (see Fig.).To
use PMMC device as a meter, two problems must be solved. First, a way must be
found to return the coil to its original position when there is no current through
the coil. Second, a method is needed to indicate the amount of coil movement.
The first problem is solved by the use of hairsprings attached to each end of the
coil as shown in Fig.
These hairsprings are not only supplying a restoring torque but also
provide an electric connection to the rotating coil. With the use of hairsprings, the
coil will return to its initial position when no current is flowing though the coil.
The springs will also resist the movement of coil when there is current through
coil. When the developing force between the magnetic fields (from permanent
magnet and electro magnet) is exactly equal to the force of the springs, the coil
rotation will stop. The coil set up is supported on jeweled bearings in order to
achieve free movement.
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1.4. Construction and Basic principle operation of Moving-iron Instruments :
We have mentioned earlier that the instruments are classified according to the principles
of operation. Furthermore, each class may be subdivided according to the nature of the
movable system and method by which the operating torque is produced. Specifically, the
electromagnetic instruments are sub-classes as (i) moving-iron instruments (ii) electrodynamic or dynamometer instruments, (iii) induction instruments. In this section, we will
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discuss briefly the basic principle of moving-iron instruments that are generally used to
measure alternating voltages and currents. In moving –iron instruments the movable
system consists of one or more pieces of specially-shaped soft iron, which are so pivoted
as to be acted upon by the magnetic field produced by the current in coil. There are two
general types of moving-iron instruments namely (i)
Repulsion (or double iron) type (ii) Attraction (or single-iron) type. The brief description
of different components of a moving-iron instrument is given below.
• Moving element: a small piece of soft iron in the form of a vane or rod
• Coil: to produce the magnetic field due to current flowing through it and also to
magnetize the iron pieces.
• In repulsion type, a fixed vane or rod is also used and magnetized with the same
polarity.
• Control torque is provided by spring or weight (gravity)
• Damping torque is normally pneumatic, the damping device consisting of an air
chamber and a moving vane attached to the instrument spindle.
• Deflecting torque produces a movement on an aluminum pointer over a graduated
scale.
1.5. Construction of Moving-iron Instruments
The deflecting torque in any moving-iron instrument is due to forces on a small
piece of magnetically ‘soft’ iron that is magnetized by a coil carrying the operating
current. In repulsion (Fig.) type moving–iron instrument consists of two cylindrical soft
iron vanes mounted within a fixed current-carrying coil. One iron vane is held fixed to
the coil frame and other is free to rotate, carrying with it the pointer shaft. Two irons lie
in the magnetic field produced by the coil that consists of only few turns if the instrument
is an ammeter or of many turns if the instrument is a voltmeter. Current in the coil
induces both vanes to become magnetized and repulsion between the similarly
magnetized vanes produces a proportional rotation. The deflecting torque is proportional
to the square of the current in the coil, making the instrument reading is a true ‘RMS’
quantity Rotation is opposed by a hairspring that produces the restoring torque. Only the
fixed coil carries load current, and it is constructed so as to withstand high transient
current. Moving iron instruments having scales that are nonlinear and somewhat crowded
in the lower range of calibration.
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1.6.
Extension of Range
Shunts are used for the extension of range of
Ammeters. So a good shunt should have the
following properties:1- The temperature coefficient of shunt should
be low
2- Resistance of shunt should not vary with time
3- They should carry current without excessive temperature rise
4- They should have thermal electromotive force with copper
* ‘Manganin’ is used for DC shunt and ‘Constantan’ as AC shunt.
Ammeter:- PMMC is used as indicating device. The current capacity of PMMC is small.
It is impractical to construct a PMMC coil, which can carry a current greater than 100
mA. Therefore a shunt is required for measurement of large currents.
Rm = Internal resistance of movement (coil) in Ω
Rsh = Resistance of shunt in Ω
Im = Ifs = Full scale deflection current of movement in
Amperes Ish = Shunt current in Amperes
I = Current to be measured in Amperes
Since the shunt resistance is in parallel with the meter movement, the voltage drop
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across shunt and movement must be same.
1.7. Multi Range Ammeter:- Let m1, m2, m3, m4 be the shunt multiplying powers for
current I1, I2, I3, I4.
Multi Range Voltmeter:
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Voltmeter & Ammeter by Moving Coil Instrument:- Same process as applied in PMMC.
Electrodynamic type Voltmeter & Ammeter:- Shunt is connected across the circuit for
ammeter and multiplier resistance is connected in series for voltmeter.
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UNIT-II
INSTRUMENT TRANSFOMERS
AND SPECIAL METERS
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2.1. Instrument Transformers:
Instrument Transformers Basics
Why instrument transformers?
In power systems, currents and voltages handled are very large.
Direct measurements are not possible with the existing equipments.
Hence it is required to step down currents and voltages with the help of instrument
transformers so that they can be measured with instruments of moderate sizes
Instrument Transformers
Transformers used in conjunction with measuring instruments for measurement purposes
are called “Instrument Transformers”.
The instrument used for the measurement of current is called a “Current Transformer” or
simply “CT”.
The transformers used for the measurement of voltage are called “Voltage transformer”
or “Potential transformer” or simply “PT”.
Fig 1. Current Transformer
Fig 2. Potential Transformer
Fig 1. indicates the current measurement by a C.T. The current being measured
passes through the primary winding and the secondary winding is connected to an
ammeter. The C.T. steps down the current to the level of ammeter.
Fig 2. shows the connection of P.T. for voltage measurement. The primary winding
is connected to the voltage being measured and the secondary winding to a
voltmeter. The P.T. steps down the voltage to the level of voltmeter.
Merits of Instrument Transformers:
1. Instruments of moderate size are used for metering i.e. 5A for current and
100 to 120 volts for voltage measurements.
2. Instrument and meters can be standardized so that there is saving in costs.
Replacement of damaged instruments is easy.
3. Single range instruments can be used to cover large current or voltage ranges,
when used with suitable multi range instrument transformers.
4. The metering circuit is isolated from the high voltage power circuits. Hence
isolation is not a problem and the safety is assured for the operators
5. There is low power consumption in metering circuit.
6. Several instruments can be operated from a single instrument
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Nominal Ratio: It is the ratio of rated primary winding current (voltage) to the rated
secondary winding current (voltage).
Turns ratio: This is defined as below
Burden of an Instrument Transformer:
The rated burden is the volt ampere loading which is permissible without errors
exceeding the particular class of accuracy.
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2.2. Current Transformer equivalent circuit:
X1 = Primary leakage reactance
R1 = Primary winding resistance
X2 = Secondary leakage reactance
Z0 = Magnetizing impedance
R2 = Secondary winding resistance
Zb = Secondary load
Note: Normally the leakage fluxes X1 and X2 can be neglected
2.3. Current transformer, simplified equivalent circuit:
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2.4. Current transformer: Phase displacement and current ratio error :
2.5. Construction of CT
Construction of Current Transformer:
Current transformers are constructed in various ways. In one method there are two
separate windings on a magnetic steel core. The primary winding consists of a few turns
of heavy wire capable of carrying the full load current while the secondary winding
consist of many turns of smaller wire with a current carrying capacity of between 5/20
amperes, dependent on the design. This is called the wound type due to its wound
primary coil.
2.6. Wound Type
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Phasor Diagram
Angle by which the reversed I2 differs in phase from the I1vector
Ideally the I2should lag the I1by 1800and hence the phase angle is ZERO
In practice this angle is < 1800due to magnetizing and loss component of the I1
Potential Transformer Basics
Potential transformers are normally connected across two lines of the circuit in which the
voltage is to be measured. Normally they will be connected L-L (line-to-line) or L-G
(line-to-ground). A typical connection is as follows:
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2.8. Phasor Diagram of Potential Transformer:
The theory of a potential transformer is the same as that of a power transformer. The
main difference is that the power loading of a P.T. is very small and consequently the
exciting current is of the same order as the secondary winding current while in a power
transformer the exciting current is a very small fraction of secondary winding load
current.
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UNIT III
Measurement of Power and Energy
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UNIT-III
POWER MEASUREMENTS
3.1. INTRODUCTION
Electric power is the rate of doing work. It is expressed in Watts. The higher units of
power used in practice include kilowatts, megawatts, etc. PWatts = VI COS φ , i.e., a power
of one watt is said to be expended when a source of one volt passes a current of one
ampere through a load resistance/ impedance of one ohm at unity power factor.
The power measurements are made with the help of a wattmeter. Wattmeter is an
indicating deflecting type of instrument used in laboratories for measurement of power in
various ranges. A wattmeter consists of two coils as shown in the schematic
representative figure 3.1
Current coil (CC): connected in series
with circuit and carries the load current.
It is designed such that it is wound with
2 to 3 turns of thick wire and hence it
has a very low resistance.
Voltage or Pressure or Potential coil (PC):
connected across the load circuit and
hence carries a current proportional to
the load current. The total load voltage
appears
across the PC. It is designed such that it is
Fig. 3.1 Wattmeter Connections
wound with several turns of thin wire and
hence it has a very high resistance.
The wattmeter can be a UPF meter or LPF meter depending on the type of the load
connected in the measuring circuit. For power measurements in AC circuits, the
wattmeter is widely adopted. In principle and construction, it is a combination of those
applicable for an ammeter and a voltmeter.
The electrical power can be of three forms:
Real power or simply, the power is the power consumed by the resistive loads on the
system. It is expressed in watts (W). This is also referred as true power, absolute power,
average power, or wattage.
Reactive power is the power consumed by the reactive loads on the system. It is
expressed in reactive volt-amperes (VAr).
Apparent power is the vector sum of the above two power components. It is expressed in
volt-amperes (VA).
Fig.3.2 The Power Triangle
Thus, it is observed from the power triangle shown in figure 3.2, that more is the
deviation of power factor from its unity value, more is the deviation of real power from
the apparent power. Also, we have
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VA2 = W2 + VAr2
(3.1)
And power factor, cos ϕ = ( Watts / VA )
(3.2)
3.2 SINGLE PHASE REAL POWER MEASUREMENTS
3.2.1 Electrodynamometer Wattmeter
An electrodynamometer wattmeter consists of two fixed coils, FA and FB and a moving
coil M as shown in figure 3.3. The fixed coils are connected in series with the load and
hence carry the load current. These fixed coils form the current coil of the wattmeter. The
moving coil is connected across the load and hence carries a current proportional to the
voltage across the load. A highly non-inductive resistance R is put in series with the
moving coil to limit the current to a small value. The moving coil forms the potential coil
of the wattmeter.
Fig. 3.3 Electrodynamometer Wattmeter
The fixed coils are wound with heavy wire of minimum number of turns. The fixed coils
embrace the moving coil. Spring control is used for movement and damping is by air.
The deflecting torque is proportional to the product of the currents in the two coils.
Theses watt meters can be used for both DC and AC measurements. Since the deflection
is proportional to the average power and the spring control torque is proportional to the
deflection, the scale is uniform. The meter is free from waveform errors. However, they
are more expensive.
3.2.2. Expression for the deflection torque:
Let iC, iP : Current in the fixed and moving coils
respectively, M : Mutual inductance between the two
coils,
θ : Steady final deflection of the instrument,
K: Spring constant,
V, I : RMS values of voltage and current in the measuring circuit and
RP : Pressure coil resistance.
Instantaneous voltage across pressure coil, v = √2 V sin wt
Instantaneous current in the pressure coil, iP = √2 V/RP sin wt = √2 IP sin wt
Instantaneous current in the current coil, iC = √2 I sin (wt-ϕ)
Instantaneous torque is given by: Ti = iC iP ( d M / d θ )
= [ √2 I sin (wt-ϕ) ] [ √2 IP sin wt ] ( d M / d θ )
T
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Average deflecting torque, Td = (1/T) ∫ Ti d wt
0
T
= (1/T) ∫ IP I [ cos ϕ - cos (2wt - ϕ ) ] ( d M / d θ ) d wt
0
= ( V I / RP ) cos ϕ ( d M / d θ )
(3.4)
Since the controlling torque, Tc=Kθ, we have at balance of the moving pointer, Td=Tc, so
that,
θ = [ V I cos ϕ / ( K RP ) ] ( d M / d θ )
= ( K’ d M / d θ ) P
(3.5)
Where K’ = K RP and P is the power consumption. Thus the deflection of the wattmeter
is found to be the direct indication of the power being consumed in the load circuit.
3.2.3 Low Power Factor Wattmeter
If an ordinary electrodynamometer wattmeter is used for measurement of power in low
power factor circuits, (PF<0.5), then the measurements would be difficult and inaccurate
since:
• The deflecting torque exerted on the moving system will be very small and
• Errors are introduced due to pressure coil inductance (which is large at LPF)
Thus, in a LPF wattmeter, special features are incorporated in a general electrodynamometer wattmeter circuit to make it suitable for use in LPF circuits as under:
(a) Pressure coil current:
The pressure coil circuit is designed to have a low value of resistance so that the current
through the pressure coil is increased to provide an increased operating torque.
(b) Compensation for pressure coil current:
On account of low power factor, the power is small and the current is high. In this
context, there are two possible connections of the potential coil of a wattmeter as shown
in figure 4.4. The connection (a) can not be used, since owing to the high load current,
there would be a high power loss in the current coil and hence the wattmeter reading
would be with a large error. If the connection (b) is used, then the power loss in the
pressure coil circuit is also included in the meter readings.
Thus it is necessary to compensate for the pressure coil current in a low power factor
wattmeter. For this, a compensating coil is used in the instrument to compensate for the
power loss in the pressure coil circuit as shown in figure 3.5.
(c) Compensation for pressure coil inductance:
At low power factor, the error caused by the pressure coil inductance is very large.
Hence, this has to be compensated, by connecting a capacitor C across a portion of the
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series resistance in the pressure coil circuit as shown in figure 3.5.
(d) Realizing a small control torque:
Low power factor wattmeters are designed to have a very small control torque so that
they can provide full scale deflection (f.s.d.) for power factor values as low as 10%. Thus,
the complete circuit of a low power factor wattmeter is as shown in figure 3.5.
3.3 REACTIVE POWER MEASUREMENTS
A single wattmeter can also be used for three phase reactive power measurements. For
example, the connection of a single wattmeter for 3-phase reactive power measurement in
a balanced three phase circuit is as shown in figure 4.6.
Fig. 4.6 Reactive power measurement circuit
The current coil of the wattmeter is inserted in one line and the potential coil is connected
across the other two lines. Thus, the voltage applied to the voltage coil is VRB= VR-VB,
where, VR and VB are the phase voltage values of lines R and B respectively, as
illustrated by the phasor diagram of figure 4.7.
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Phasor diagram for reactive power measurements
The reading of the wattmeter, W3ph for the connection shown in figure 3.6 can
be
obtained based on the phasor diagram of figure 34.7, as follows:
Wattmeter reading,
Wph = Iy VRB
= Iy VL cos (90+Ø)
= - √3 Vph Iph sin Ø
= - √3 (Reactive power per phase)
Thus, the three phase power, W3ph is given by,
W3ph = (VArs/phase)
= 3 [Wph /- √3]
= - √3 (wattmeter reading)
(3.6)
(3.7)
3.4 THREE PHASE REAL POWER MEASUREMENTS
The three phase real power is given by,
P3ph= 3 Vph Iph cos Ø
or
P3ph= √3 VL IL cos Ø
(3.8)
The three phase power can be measured by using either one wattmeter, two wattmeters or
three wattmeters in the measuring circuit. Of these, the two wattmeter method is widely
used for the obvious advantages of measurements involved in it as discussed below.
3.4.1 Single Wattmeter Method
Here only one wattmeter is used for measurement of three phase power. For circuits with
the balanced loads, we have: W3ph=3(wattmeter reading). For circuits with the
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unbalanced loads, we have: W3ph=sum of the three readings obtained separately by
connecting wattmeter in each of the three phases. If the neutral point is not available (3
phase 3 wire circuits) then an artificial neutral is created for wattmeter connection
purposes. Instead three wattmeters can be connected simultaneously to measure the three
phase power. However, this involves more number of meters to be used for
measurements and hence is not preferred in practice. Instead, the three phase power can
be easily measured by using only two wattmeters, as discussed next.
3.4.2 Two Wattmeter Method
The circuit diagram for two wattmeter method of measurement of three phase real power
is as shown in the figure 34.7. The current coil of the wattmeters W1 and W2 are inserted
respectively in R and Y phases. The potential coils of the two wattmeters are joined
together to phase B, the third phase. Thus, the voltage applied to the voltage coil of the
meter, W1 is VRB= VR-VB, while the voltage applied to the voltage coil of the meter, W2
is VYB=VY-VB, where, VR, VB and VC are the phase voltage values of lines R, Y and B
respectively, as illustrated by the phasor diagram of figure 3.8. Thus, the reading of the
two wattmeters can be obtained based on the phasor diagram of figure 4.8, as follows:
Hence,
And
So that then,
W1 = IR VRB
= IL VL cos (30 - Ø)
W2 = IY VYB
= IL VL cos (30 + Ø)
W1+W2 = √3 VL IL cos Ø = P3ph
W1-W2 = VL IL sin Ø
(3.9)
(3.10)
(3.11)
(3.12)
Tan Ø = √3 [W1-W2]/ [W1+W2]
(3.13)
Where Ø is the lagging PF angle of the load. It is to be noted that the equations (3.11) and
(3.12) get exchanged if the load is considered to be of leading PF.
Two wattmeter method of 3-phase power measurement
Phasor diagram for real power measurements
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The readings of the two wattmeters used for real power measurements in three phase
circuits as above vary with the load power factor as described in the table 3.1.
Variation of wattmeter readings with load PF (lag)
PF angle
φ (lag)
00
300
600
>600
W1
W2
W3ph=W1+W2
PF
cos φ VLILcos(30-φ) VLILcos(30+φ) √3VLILcosφ
UPF
2W1 or 2W2
√3/2 VLIL
√3/2 VLIL
0.866
VLIL
VLIL/2
1.5W1 or 3W2
√3/2 VLIL
0.5
ZERO
W1 alone
<0.5
W1
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W2 reads
negative
W1+(-W2)
Remarks
Gen. Case (always W1≥W2)
W1=W2
W2=W1/2
W2 reads zero
For taking readings, the PC
or
CC connection of W2 should
be reversed) (LPF case)
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UNIT-IV
Potentiometers
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4.1. Potentiometers
Before the introduction of the moving coil galvanometer, potentiometers were used
in measuring voltage, hence the '-meter' part of their name
Today this method is still important in standards work
The null-balance principle of measurement is also used in other areas of electronics
The potentiometer used for measurement is a type of bridge circuit for measuring
voltages by comparison between a small fraction of the voltage which could be
precisely measured, then balancing the two circuits to get null current flow which
could be precisely measured
Types
Main Classification
Constant current potentiometer
Constant resistance potentiometer
Microvolt potentiometer
Thermocouple potentiometer
Other
DC
AC
4.2. Construction of potentiometers
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UNIT-V
Resistance Measurement
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5.1 INTRODUCTION
A resistance is a parameter which opposes the flow of current in a closed electrical
network. It is expressed in ohms, milliohms, kilo-ohms, etc. as per the ohmic principle, it is
given by;
R = V/I; with temperature remaining constant.
(1)
If temperature is not a constant entity, then the resistance, R2 at t20C is given by;
R2 = R1 [1+ α (t2-t1)] (2) Where t= t1-t2, the rise in temperature from t10C to t20C, α
is the temperature coefficient of resistance and R1 is the temperature at t10C.
Resistance can also be expressed in terms of its physical dimensions as under:
R = ρl/A (3) Where, l is the length, A is the cross sectional area and ρ is the specific
resistance or resistivity of the material of resistor under measurement.
To realize a good resistor, its material should have the following properties:
• Permanency of its value with time.
• Low temperature coefficient of resistance
• High resistivity so that the size is smaller
• Resistance to oxidation, corrosion, moisture effects, etc.
• Low thermo electric emf against copper, etc.
No single material can be expected to satisfy all the above requirements together. Hence, in
practice, a material is chosen for a given resistor based on its suitability for a particular
application.
5.2 CLASSIFICATION OF RESISTANCES
For the purposes of measurements, the resistances are classified into three major groups
based on their numerical range of values as under:
• Low resistance (0 to 1 ohm)
• Medium resistance (1 to 100 kilo-ohm) and
• High resistance (>100 kilo-ohm)
Accordingly, the resistances can be measured by various ways, depending on their range of
values, as under:
1. Low resistance (0 to 1 ohm): AV Method, Kelvin Double Bridge, potentiometer,
doctor ohmmeter, etc.
2. Medium resistance (1 to 100 kilo-ohm): AV method, wheat stone’s bridge,
substitution method, etc.
3. High resistance (>100 kilo-ohm): AV method, Fall of potential method, Megger,
loss of charge method, substitution method, bridge method, etc.
5.3.
MEASUREMENT OF MEDIUM RESISTANCES
(a) AV Method
Here an ammeter and a voltmeter are used respectively in series and in parallel with the
resistor under measurement. Various trial readings are obtained for different current
values. Resistances are obtained for each trial as per ohmic-principle, using equation (1).
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The average value of all the trails will give the measured value of resistance.
This method is also reefed as the potential drop method or VA method. Here, the meter
ranges are to be chosen carefully based on the circuit conditions and the resistance value to
be obtained. This method suffers from the connection errors.
(b) Wheatstone’s bridge
This bridge circuit has four resistive arms: arm-AB and arm-BC the ration arms with
resistances R1 and R2, arm-AD with the unknown resistance R3 and arm CD with a
standard known variable resistance R4, as shown in figure-1 below. The supply is fed
across the arm-AC and the arm-BD contains a galvanometer used as a detector connected
across it.
Fig.1 Wheatstone bridge
The bridge is said to be balanced when the galvanometer current is zero. This is called as
the position of null reading in the galvanometer, which is used here as a null detector of the
bridge under the balanced conditions.
Thus under the balance conditions of the bridge, we have,
Ig = 0; VAB = VAD and VBC = VCD, so that,
I1R1 = I2R3 and I1R2 = I2R4
(4)
Solving further, we get, R1/R2 = R3/ R4 and R1R4 = R2R3
(5)
This is the balance equation of the bridge.
Thus, unknown resistance, R3 = R1R4/ R2 ohms.
(6)
5.4.
Expression for Galvanometer current
In a WS bridge PQRS as shown in figure 2, below, under the unbalanced conditions of the
bridge, the current flowing in the galvanometer is given by:
Ig = ETh/ [RTh+Rg]
(7)
Where,
RTh = {PQ/(P+Q) + RS/(R+S)}
ETh = E {P/(P+Q) - R/(R+S)}
(8)
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Fig.2 Wheatstone bridge PQRS
5.5.
Errors in WS bridge measurements
(i) Limiting errors: In a WS bridge PQRS, the percentage limiting error in the measurand
resistance, R is equal to the sum of the percentage limiting errors in the bridge arm
resistances P, Q and S.
(ii) Errors due to heating of elements in the bridge arms:
Rt = R0 [1+ αt]; P = I2R Watts; Heat= I2Rt Joules
The I2R loss occurring in the resistors of each arm might tend to increase the temperature,
which in turn can result in a change in the resistance value, different from the normal
value.
(iii) Errors due to the effect of the connecting wires and lead resistors: the connecting
lead wire resistance will affect the value of the unknown resistance, especially when it is a
low resistance value. Thus, the connecting lead wire resistances have to be accounted for
while measuring a low resistance.
(iv) Contact resistance errors: The contact resistance of the leads also affects the value of
the measurand resistance, just as in point (iii) above. This resistance value depends on the
cleanliness of the contact surfaces and the pressure applied to the circuit.
1.3.3 Limitations of WS bridge
The WS bridge method is used for measurement of resistances that are numerically in the
range of a few ohms to several kilo-ohms. The upper limit is set by the reduction in
sensitivity to unbalance caused by the high resistances, as per the equation (7).
1.3.4 Sensitivity of WS bridge
In a WS bridge PQRS as in figure 2, with the resistance S in an arm changed to S+ S, the
bridge becomes unbalanced to the extent of the resistance change thus brought about. This
is also referred as the unbalanced operation of the WS bridge. Under such circumstances,
with Sv as the voltage sensitivity of the galvanometer, we have,
θ = Sv e = Sv [ES R/(R+S)2]
Galvanometer Deflection,
(10)
S
=
θ/(
R/R)
Bridge Senstivity,
(11)
B
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Rearranging, we get,
SB = Sve/{(R/S)+2+(S/R)}
(12)
Thus, the WS bridge sensitivity is high when (R/S)=1. It decreases as the ratio (R/S)
becomes either larger or smaller than unity value. This also causes a reduction in accuracy
with which the bridge is balanced.
1.4 MEASUREMENT OF LOW RESISTANCES
Of the various methods used for measurement of low resistances, the Kelvin’s Double
Bridge (KDB) is the most widely used method. Here, the bridge is designed as an
improved or modified WS Bridge with the effect of contact and connecting lead wire
resistances considered. The circuit arrangement for KDB is as shown in figure 3 below. R
is the low resistance to be measured, S is the standard low resistance, r is the very small
link resistance between R and S, called the yoke resistance, P,Q,p,q are the non inductive
resistances, one set of which, Pp or Qq is variable. The bridge is supplied with a battery E
and a regulating resistance RC. A galvanometer of internal resistance Rg, connected as
shown, is used as a null detector.
Fig.3 Kelvin’s Double Bridge
1.4.1 Balance Equation of KDB
The balance equation of KDB can be derived as under. Using star-delta conversion
principle, the KDB circuit of figure 3 can be simplified as shown in figure-4 below.
(a)
(b)
(c)
Fig. 4 Simplified circuits of KDB
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It is thus observed that the KDB has now become a simplified WS bridge-ABCD as shown
in figure 4(c). It is to be noted that the yoke resistance r1 is ineffective now, since it is in
the zero current branch, in series with the galvanometer. Thus, under balance conditions,
we have,
P (S+r3) = Q (R+r2)
(13)
Where,
r1 = pq / (p+q+r)
r2 = pr / (p+q+r)
r3 = qr / (p+q+r)
(14)
Using (13) and (14), we get,
PS + Pqr / (p+q+r) = QR + Qpr / (p+q+r)
(15)
Simplifying (15), we get the final balance equation of the Kelvin Double Bridge as:
R = [P/Q] S + [qr / (p+q+r)] [(P/Q) – (p/q)]
(16)
1.4.2 Significance of balance equation of KDB
From the balance equation (16), it can be observed that,
• If the yoke resistance is negligible, (r=0), then R = [P/Q]S, as per the WS bridge
balance equation.
• If the ratio arm resistances are equal, i.e., [(P/Q) = (p/q)], then again, R=[P/Q]S, as
per WS bridge principle.
1.4.3 Unbalanced KDB
Further, if the KDB is unbalanced due to any change in resistances R or S (by R or S), then
the unbalanced galvanometer current during the unbalanced conditions is given by:
Ig = I. S [P/(P+Q)]/ {Rg+ pq/(p+q)+ PQ/(P+Q)}
(17)
1.4 MEASUREMENT OF HIGH RESISTANCES
Of the various methods used for measurement of high resistances, the fall of potential
method is widely used. Also, very high resistances of the order of mega-ohms can be
measured by using an instrument called MEGGER, also called as the insulation resistance
tester. It is used as a high resistance measuring meter as also a tester for testing the earth
resistances.
1.5 SOLVED PROBLEMS
1
Calculate the current through the galvanometer for the Wheatstone bridge circuit
shown below in figure P1.
R1 = 2 k
R2 = 4 k
R3 = 7 k
R4 = 20 k
Rg = 300
V = 8.0 V
Fig. P1
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Solution:
The Thevenin’s equivalent circuit is used to find Ig. To find ETh, remove Rg as
shown in figure P1(a).
VTh = VCD = I2x4 k – I1x2k
I1= 8 / [ (2+7)103] = 8.888(10-4) A
I2 = 8 / [ (4+20)103] = 3.333(10-4) A
Thus, VTh = 3.333(10-4) x4(103)– 8.888(10-4) x2(103)
V
= - 0.4444 V, i.e., point C is negative.
Fig. P1(a)
To find RTh, short the battery as shown in figure P1(b). The points A and B are
shorted and can be represented by a single point as shown in figure P1(c).
RTh = [2(7)/(2+7)]
+ [4(20)/(4+20)]
= [1.555 + 3.333] k
= 4.888 k
Fig. P1(b)
Fig. P1(c)
Ig = VTh / [Rg+RTh]
= 0.4444 / [4.888x103+300]
= 85.65 µA - from D to C
as shown in
figure P1(d)
aside
Fig. P1(d)
2
A KDB is balanced with the following constants: outer ratio arms: 100 and 1000.
Ohms, inner ratio arms: 99.92 and 1000.6 ohms, link resistance: 0.1 ohm, standard
resistance: 3.77 milliohms, find the unknown resistance.
Solution:
The KDB circuit is as shown in figure P2.
P = 100 ohms Q
= 1000 ohms p =
99.92 ohms q =
1000.6 ohms r =
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0.1 ohm
S = 3.77(10-3) ohms
= 0.00377 ohms
Fig. P2
The unknown resistance R is given by:
R = [P/Q] S + [qr / (p+q+r)] [(P/Q) – (p/q)]
By substituting the specified values, we get,
R=
[100/1000]0.00377+[1000.6(0.1)/(99.92+1000.6+0.1)][(100/1000)–
(99.92/1000.6)]
= 0.382712 milliohms.
3
The ratio arms of a KDB are 1000 ohms each. The galvanometer has a resistance of
1000 ohms and a sensitivity of 0.003 µA/mm. A current of 10 A is passed by a 2.2
V battery in series with a rheostat. The yoke resistance is negligible. The standard
resistance is 0.1 ohm. The galvanometer gave a deflection of 25 mm. by how much
the unknown differs in resistance from the standard one.
Solution :
The KDB circuit with the specified values of all the parameters of the bridge is as
shown in figure P3.
Fig.P3
I = kθ = 0.003µA/mm (25 mm) = 0.075 µA
R = (P/Q)S = [1000/1000] (0.1) = 0.1 ohm.
Ig = I. R [P/(P+Q)]/ {Rg+ pq/(p+q)+ PQ/(P+Q)}
By substituting the
specified values, we get,
0.075(10-6) = 10 R [1/2]/[1000+500+500]
Thus, solving for R, we ger,
R = 30 µ
4
Given that, in a KDB circuit, with the usual notations, P=Q=p=q=1000 ohms,
E=100V, RC=5 ohms, Rg=500 ohms and the bridge is balanced with S=0.001 ohm,
what is the approximate value of current through S, at balance? Also find the value
of galvanometer current Ig, if S is changed by 1% of its value at balance.
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Solution :
Here, it is assumed that the yoke resistance is zero, i.e., r=0, so that
then, R=(P/Q)S = 0.1 ohm.
(i) I ≈ E/{R+Rc+r+S}
= 100/ [0.001+5+0+0.001] = 20 A
(ii) S = (0.01) S = 0.01(0.001) = 10 µ
Thus,
Ig = Ig = I. S [P/(P+Q)]/ {Rg+ pq/(p+q)+ PQ/(P+Q)} = 0.0667 µA
5
The ratio arms of a KDB are 1000 each, Rg=500 , k=200 mm/µA, R=0.1002 and
S=0.1 . A DC of 10 A is passed through R & S from a 2.2 V battery in series with a
rheostat. The link resistance is negligible. (i) Find the galvanometer deflection (ii)
Find the resistance unbalance to produce the deflection of 1 mm and (iii) Obtain the
total internal resistance of the battery circuit.
Solution :
(i) R = (P/Q)S = 0.1 ohm
Thus, R = 0.1002-0.1 = 200 µ
Hence,
Ig = I. S [P/(P+Q)]/ {Rg+ pq/(p+q)+ PQ/(P+Q)} = 1.6667 µA
And θg = k Ig = 333.33 mm
(ii) θg = 1 mm, -6Ig = θ/k = 1/200 = 0.005 µA
Thus, 0.005(10 ) = 10 R(0.5)/[(500+50+50]
Simplifying and solving, we get, R = 0.6 µ
(iii) I ≈ E/{R+Rc+r+S}
10 = 2.2 / [0.1002+RC+0.1+0]
Solving, we get, RC = 0.0198 ohm.
6
A KDB is as follows: P=p=10000 , Q=q=1000 , R=0.0997 , S=0.01 , Rg=1000 ,
r=0.01 and k=0.005 µA/mm. A battery of 2.25 Volts and internal resistance 0.0053
is connected across the bridge circuit. Find the galvanometer deflection.
Solution :
R = (P/Q)S = 0.1 ohm (since P/Q = p/q)
Thus, R = Rtrue – Rmeasured = 0.0997 – 0.1 = 300 µ
I ≈ E/{R+Rc+r+S}
= 2.25 / [0.0997+0.0053+0.01+0.01]
= 18 A
Ig = I. S [P/(P+Q)]/ {Rg+ pq/(p+q)+ PQ/(P+Q)} = 1.742 µA
And
θg = Ig/k = 91.742 µA)/(0.005 µA/mm)
= 348.4 mm.
1.6 EXERCISES
Standard text books listed in the syllabus are to be referred for the worked examples on the
topics covered so far for better understanding of the problem solving techniques on
resistance measurements.
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UNIT-VI
AC Bridges
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UNIT-VI
5.1 General Form of the AC Bridge
5.2 Maxwell Bridge
The Maxwell bridge measures an unknown inductance in terms of a known
capacitance.
The maxwell bridge is limited to the measurement of medium-Q coils (1<Q<10).
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5.3 Hay Bridge
 The Hay circuit is more convenient for measuring high-Q coils
 Hay bridge for inductance measurements
 for Q>10 : Lx= R2R3C
5.4 Schering Bridge:
The Schering bridge, one of the most important bridges, is used extensively for the
measurement of capacitors.
Schering bridge for measurement of capacitance
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5.5 Wien Bridge :
Applications :
– Frequency measureent
– Notch filter
– Frequency-determining element
Frequency measurement with the Wien bridge
5.6 Anderson Bridge:
 In the Anderson Bridge the unknown inductance is measured in terms of a known
capacitance and resistance.
 this method is capable of precise measurements of inductance over a wide range of
values from a few micro-henrys to several henrys and is the best bridge method
5.7 Capacitance Bridge:
We will consider only De Sauty bridge method of comparing two capacitances the bridge has
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maximum sensitivity when C2 = C3.
 The simplicity of this method is offset by the impossibility of obtaining a perfect
balance if both the capacitors are not free from the dielectric loss.
 A perfect balance can only be obtained if air capacitors are used. as shown in fig
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